Problem 76 - Prealgebra

Question

Add or subtract as indicated. $$\frac{1}{3}+\frac{4}{12}$$

Step-by-Step Solution

Answer

\( \frac{2}{3} \)

Step 1: Identify common denominators

First, identify the denominators of the fractions. The denominators for the fractions \( \frac{1}{3} \) and \( \frac{4}{12} \) are 3 and 12, respectively.

Step 2: Find the least common denominator

Determine the least common denominator (LCD) for the fractions. The smallest number that both 3 and 12 can divide without a remainder is 12. Therefore, the LCD is 12.

Step 3: Convert fractions to common denominators

Convert \( \frac{1}{3} \) so it has the denominator of 12. Multiply both the numerator and denominator by 4 to get \( \frac{4}{12} \). The second fraction, \( \frac{4}{12} \), already has the denominator of 12, so no changes are needed.

Step 4: Add the numerators

Now, add the numerators of the fractions. The problem simplifies to \( \frac{4}{12} + \frac{4}{12} \). Add the numerators: \( 4 + 4 = 8 \). So, \( \frac{4}{12} + \frac{4}{12} = \frac{8}{12} \).

Step 5: Simplify the fraction

Simplify \( \frac{8}{12} \) by finding the greatest common divisor (GCD) of 8 and 12, which is 4. Divide both the numerator and the denominator by 4. Thus, \( \frac{8}{12} \) simplifies to \( \frac{2}{3} \).

Key Concepts

Least Common Denominator

When adding or subtracting fractions, each fraction must have the same denominator. This common denominator is known as the Least Common Denominator (LCD). The LCD is simply the smallest number that all denominators in the problem can divide into evenly.

To determine the LCD, you should look for the least common multiple of the denominators. For example, with fractions \( \frac{1}{3} \) and \( \frac{4}{12} \), the denominators are 3 and 12. The smallest number that both 3 and 12 can divide into evenly is 12, which becomes our LCD.

Finding the LCD is essential because it allows us to combine the fractions by making sure they share the same denominator. This step makes adding fractions straightforward because it aligns their size relative to each other.

Simplifying Fractions

After adding fractions, you may end up with a fraction that can be simplified, meaning its numerator and denominator can both be divided by the same number. Simplifying a fraction makes it easier to understand and work with.

Once you have completed the addition or subtraction, examine the resulting fraction. If both the numerator and denominator are divisible by the same number, reduce the fraction by dividing them both by this common number.

For example, the fraction \( \frac{8}{12} \) can be simplified because both 8 and 12 can be divided by the same number - 4. Dividing gives \( \frac{2}{3} \), which is the simplest form of \( \frac{8}{12} \). Always ensure your final answer is in its simplest form to make it clear and concise.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is key to simplifying fractions, as it is the largest number that divides both the numerator and denominator without a remainder. Identifying the GCD is the first step in reducing a fraction to its simplest form.

For example, when simplifying the fraction \( \frac{8}{12} \), you first identify that the GCD of 8 and 12 is 4. Divide both the numerator (8) and the denominator (12) by the GCD. This process reduces the fraction to \( \frac{2}{3} \).

Understanding how to determine the GCD can help you work with fractions more effectively, making it easier to see the simplest form of the fraction. Often, this involves listing the factors of each number, then identifying the largest factor they share. This simplification process not only makes your final answer easier to interpret but also aligns more closely with how numbers are typically used in real-world scenarios.